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(use (a)). Deduce that CT(XI, ^2) = a(x*  X<) and that

Deduce from this inequality that, if %lt #2 , Xz are tnree characters of A, we have


and hence that the relation a(^', x") < 1 between characters ^', x" of A is an equiva-
lence relation on X(A). The equivalence classes for this relation are called Gleason
parts of X(A).

(c) Let u e A be such that p(u) ^ 1, and let ^i, X* ^e two characters of A, such that
Xi(w) = 0. Show that there exists a complex number A such that |A| < 1 and such that,
if we put v = (A  w)(l  Aw)~l e A, then xi(v) = A and x2(t>) = A. Deduce that

}<    4|XI(,)

*2   ; =

(d)   If xi, X2 are two characters of A, put

T(xi,x2) =    SUP    \

ue A, /)(u)< 1

Show that

(use (c)). The equivalence relation defined in (b) is therefore equivalent to r(x', x") < 2.
(e) Let we A be such that ^x(u) = ^ ^or a11 Xe*(A). Show that, for any two
characters Xi > X2 > we nave

(Apply (*) to the element e~ru, where / > 0, and let t tend to 0.)

19.   With the hypotheses of Problem 8, let xi, X* ^e two characters of B.

(a)    Let ftj.,/^2 be representative measures for %i,X2 respectively (Problem 9).
If cr(xi, ^2) = 1 (Problem 18), show that ^ and ju,2 are disjoint (13.18). (Write ft2 ^
h - fji! + v where v is disjoint from /xj, and by majorizing |^2(w)  XiWI for u e B

and ||u|| ^1, show that the hypothesis implies that I (h + 1) d^ g I \h  1| flf/x!.)

(b)    If <T(XI, x^)< 1> show that there exist representative measures pi for ^i and
fju2 for X2 such that

(Put c== (1  o-(^1? ^2)(1 + o-(xr, Xz))"1- Use Problem 18(e) and the Hahn-Banach(/x) and conversely (Problem 15(e)). A function